Theorem vtoclOLD 3559

Description: Obsolete version of vtocl 3558 as of 20-Jun-2025. (Contributed by NM, 30-Aug-1993.) Remove dependency on ax-10 2139. (Revised by BJ, 29-Nov-2020.) (Proof shortened by SN, 20-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vtocl.1	⊢ 𝐴 ∈ V
vtocl.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
vtocl.3	⊢ 𝜑

Assertion

Ref	Expression
vtoclOLD	⊢ 𝜓

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem vtoclOLD

Step	Hyp	Ref	Expression
1		vtocl.3	. . 3 ⊢ 𝜑
2		vtocl.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	mpbii 233	. 2 ⊢ (𝑥 = 𝐴 → 𝜓)
4		vtocl.1	. . 3 ⊢ 𝐴 ∈ V
5	4	isseti 3496	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
6	3, 5	exlimiiv 1929	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  Vcvv 3478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-clel 2814

This theorem is referenced by: (None)